Question

How do you simplify $\sqrt{45{{x}^{2}}}$ ?

Answer 1

Hint: The given expression contains a constant along with a variable under a square root. To solve this, try simplifying the constants under the root Firstly, and then apply the Multiplication Law to dissociate the terms. Evaluate further to get a single term or a more simplified term.

Complete step by step solution:
The given expression is, $\sqrt{45{{x}^{2}}}$
First let us simplify the constant under the root.
For this we shall write the constant which is $45\;$ in factorial form.
Upon expanding the constant, we get,
$\Rightarrow \sqrt{3\times 3\times 5\times {{x}^{2}}}$
According to one of the Laws of radicals which is the multiplication Law of Radicals,
States that $\sqrt[n]{{ab}} = \sqrt[n]{a} \times \sqrt[n]{b}$only if $a,b > 0$.
By using the above reference our expression can be written as,
$\Rightarrow \sqrt{3\times 3\times 5}\times \sqrt{{{x}^{2}}}$
We can write the respective constant in their square-form as below.
$\Rightarrow \sqrt{{{3}^{2}}}\times \sqrt{5}\times \sqrt{{{x}^{2}}}$
On further simplification, the square and square root gets canceled.
This applies for both constant as well as the variables.
$\Rightarrow 3\times \sqrt{5}\times x$
Further, we can rearrange the terms to get,
$\Rightarrow 3x\sqrt{5}$
Hence, $\sqrt{45{{x}^{2}}}$ on evaluation we get $3x\sqrt{5}$

Note: The expression $\sqrt[n]{a}$, n is called index,$\sqrt{{}}$is called radical, and $a$ is called the radicand.$\sqrt[n]{a}={{a}^{\frac{1}{n}}}$The left side of the equation is known as radical form and the right side is exponential form. A Radical represents a fractional exponent in which the numerator and the denominator of the radical contains the power of the base and the index of the radical, respectively. Memorizing all the Laws of radicals will help to solve any type of same model questions easily. It is important to reduce a radical to its simplest form using the Laws of Radicals for multiplication, division, raising an exponent to an exponent, and taking a radical of a radical makes the simplification process for radicals much easier.